Re: abundance of irrationals!)

imaginatorium@xxxxxxxxxxxxx said:
>
> aeo6 Tony Orlow wrote:
> > imaginatorium@xxxxxxxxxxxxx said:
> > >
> > > aeo6 Tony Orlow wrote:
> > > > mueckenh@xxxxxxxxxxxxxxxxx said:
> > > > >
> > > > >
> > > > > Remember: Every set of even numbers contains numbers which are
> > > larger
> > > > > tan the cardinality of that set. This is valid for *every* set
> of
> > > > > finite numbers. There is no reasonable arguing claiming that
> this
> > > > > situation should change in case of an "infinite" set.
> > >
> > > Yes there is - it's the error contained in the "salient point"
> below.
> > >
> > > Also some
> > > > > elements of an infinite set of even numbers would necessarily
> > > surpass
> > > > > the cardinality.
> > > > Yes. The **salient point** is that the set size can never be
> LARGER than
> > > the
> > > > largest element.
> > >
> > > Absolutely! The set size can never be LARGER than the largest
> element.
> > > Shout it from the rooftops! If you can find the largest element,
> the
> > > set size cannot be larger. Great! Do we all agree now??
> > >
> > > Uh-huh. Problem. What happens when there is NO largest element?
> Then
> > > the set size cannot be larger than, er, um, well, come in Houston,
> we
> > > don't seem to have a largest element here.
> > >
> > > Brian Chandler
> > > http://imaginatorium.org
> > >
> > >
> > Brian I already responded to that, you fool.
>
> Insults are very unproductive. You do not help by throwing words like
> "moron" at people like Randy Poe, who show unimaginable patience in
> pointing out your mistakes over and over again. Anyway...

Excuse me, but he did not even read what I wrote. I keep having to repeat
myself as the cantorians change the subject and refuse to follow the logic.
People want top harp on the fact that there is no largest finite? Fine. I
changed it to "some element must be at least as large as the set size." When
people ask the same dumb questions and don't pay attention to answers, or say
you said something you didn't, just to avoid the point, what should I say?
>
> > You claim an infinite set
> > containing only finite values. You claim the argument fails because
> there is no
> > largest element, but whatever it is, it's finite. Well, if all values
> in the
> > set are finite, then they are all less than infinite, **so** the set
> size cannot be
> > infinite. QED
>
> Why "so" (my asterisked emphasis)? The set consists entirely of finite
> values. The fact that there is an unlimited number of them does not *of
> itself* imply that any of these values is itself finite - as you
> yourself agree, in the case of (e.g.) the rationals in [0, 1]. So why
> do you think it does in this case? Well, look at _your own_ argument.
> The "salient point" you say is something to do with the largest
> element; but then when we point out that no largest element exists,
> somehow this "salient point" doesn't matter any more?

When you wrote this, I had already posted two proofs. Why don't you comment on
them instead of other people's comments or my comments to them? I have
explained the difference between [0,1] in the reals and the set of naturals
clearly. Did you miss that or are you repeating the question to be annoying? Do
the reals in [0,1] differ by a constant finite amount? No? An infinitesimal
amount? Well, then, it's no wonder you can squeeze an infinite number of them
in a finite space. The naturals, on the other hand, each extend the set values
by 1, a constant finite increase, an infinity of which will result in an
infinite increase in values, to infinite values.
>
> Anyway, it's not surprising you think there's something wrong with set
> theory, given your extraordinary misunderstanding of it.

I don't misunderstand it. I disagree with it and find its method and
conclusions to be unacceptable for reasons of consistency and logic.

> One of the
> most basic things set theory allows us to do is to consider the element
> of all elements (within some universe of discourse) having a particular
> property.
Set of all sets, you mean? No wait, let me try that again, in typical math-
group-speak:

The element of all elements?? What does that mean? Obviously you need to take a
course in math, or English, or maybe none of that will help. It's so sad how
confused you are. There are no elements in elements, unless the element is a
set in a set of other sets. Give up math if you can't even keep that straight.

> When we say "The set of all x having property P", we mean
> (obviously) that every single element x of this set does have the
> property P. Is this clear to you?
Sure. It doesn't mean the statement is true.

> If so, when I say "The set of all integers x, each x being a finite
> integer", why do you claim that by some mysterious process at least one
> other element y, not having the property of y being a finite integer
> will somehow creep in?

All I have been saying, over and over and over again, is that if the values are
all finite, then the set size is necessarily finite, because with the addition
of an infinite number of values, each 1 greater than the last, the values of
the elements will achieve infinite values. Define your set as you like, just
keep in mind the implications of your definition.

>
> Several people have asked you - over and over again - well, consider
> the set with all the 'naughty' elements removed. In terms of your
> "integers" (actually the 10-adics) including "numbers" with an
> unlimited number of nonzero digits, do you think that if you consider
> any one of these it must either have or not have an unlimited number of
> nonzero digits? If so, how many of then do *not* have an unlimited
> number of nonzero digits?

Is that your definition of a finite number? Its incorrect. 100000..... has only
one non-zero digit and is infinite. I had made a suggestion before that a zero
in the leftmost digit means its finite, but I see now that's not right either.
A finite number is one where all nonzero digits are a finite number of digits
from the rightmost digit.

>
> > You certainly draw a lot of conclusions about a set which you cannot
> enumerate.
>
> Which set (precisely) is that?

Gee, which set have we been discussing? I seem to remember one where we can
count forever and never stop, but the values are all finite, but there are an
infinite number of them, although every time you count and add the next higher
value to the set you also increase the set size by one, so even though they
grow at precisely the same rate, the set size achieves infinity, but that range
of values in the set mysteriously never achieves infinity, because natural
numbers are for counting and we can't count to infinity, even though we can
keep adding elements until we achieve a count of elements, uh, no, I mean set
size, which is infinite. What did you call this shredded monkeys's fist? Oh
yeah. The "naturals".
>
> Brian Chandler
> http://imaginatorium.org
>
>

--
Smiles,

Tony
.

Relevant Pages

  • Re: infinity
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  • Re: infinity
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  • Re: infinity
    ... > aeo6 Tony Orlow wrote: ... >> Value range is well defined as the upper bound for the maximum difference ... any set of consecutive naturals has a value ... the entire set because induction doesn't cover infinite cases, ...
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  • Re: abundance of irrationals!)
    ... > aeo6 Tony Orlow wrote: ... >> some infinite set of symbols to represent them. ... > representation of the naturals? ... string, determines, along with the number of symbols, how many unique strings ...
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  • Re: abundance of irrationals!)
    ... >> aeo6 Tony Orlow wrote: ... >>> You claim an infinite set ... >> unlimited number of nonzero digits, do you think that if you ...
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